Download Full text

FIBONACCI IDENTITIES AND GRAPH COLORINGS
CHRISTOPHER J. HILLAR AND TROELS WINDFELDT
Abstract. We generalize both the Fibonacci and Lucas numbers to the context of graph
colorings, and prove some identities involving these numbers. As a corollary we obtain new
proofs of some known identities involving Fibonacci numbers such as
Fr+s+t = Fr+1 Fs+1 Ft+1 + Fr Fs Ft − Fr−1 Fs−1 Ft−1 .
1. Introduction
In graph theory, it is natural to study vertex colorings, and more specifically, those colorings in which adjacent vertices have different colors. In this case, the number of such
colorings of a graph G is encoded by the chromatic polynomial of G. This object can be
computed using the method of “deletion and contraction”, which involves the recursive combination of chromatic polynomials for smaller graphs. The purpose of this note is to show
how the Fibonacci and Lucas numbers (and other integer recurrences) arise naturally in this
context, and in particular, how identities among these numbers can be generated from the
different choices for decomposing a graph into smaller pieces.
We first introduce some notation. Let G be a undirected graph (possibly containing loops
and multiple edges) with vertices V = {1, . . . , n} and edges E. Given nonnegative integers
k and `, a (k, `)-coloring of G is a map
ϕ : V → {c1 , . . . , ck+` },
in which {c1 , . . . , ck+` } is a fixed set of k + ` “colors”. The map ϕ is called proper if whenever
i is adjacent to j and ϕ(i), ϕ(j) ∈ {c1 , . . . , ck }, we have ϕ(i) 6= ϕ(j). Otherwise, we say that
the map ϕ is improper. In somewhat looser terminology, one can think of {ck+1 , . . . , ck+` }
as coloring “wildcards”.
Let χG (x, y) be a function such that χG (k, `) is the number of proper (k, `)-colorings of
G. This object was introduced by the authors of [2] and can be given as a polynomial in
x and y (see Lemma 1.1). It simultaneously generalizes the chromatic, independence, and
matching polynomials of G. For instance, χG (x, 0) is the usual chromatic polynomial while
χG (x, 1) is the independence polynomial for G (see [2] for more details).
We next state a simple rule that enables one to calculate the polynomial χG (x, y) recursively. In what follows, G\e denotes the graph obtained by removing the edge e from G, and
for a subgraph H of G, the graph G\H is gotten from G by removing H and all the edges
of G that are adjacent to vertices of H. Additionally, the contraction of an edge e in G is
the graph G/e obtained by removing e and identifying as equal the two vertices sharing this
edge.
The first author is supported under a National Science Foundation Postdoctoral Fellowship.
220
VOLUME 46/47, NUMBER 3
FIBONACCI IDENTITIES AND GRAPH COLORINGS
Lemma 1.1. Let e be an edge in G, and let v be the vertex to which e contracts in G/e.
Then,
χG (x, y) = χG\e (x, y) − χG/e (x, y) + y · χ(G/e)\v (x, y).
(1.1)
Proof. The number of proper (k, `)-colorings of G\e which have distinct colors for the vertices
sharing edge e is given by χG\e (k, `) − χG/e (k, `); these colorings are also proper for G. The
remaining proper (k, `)-colorings of G are precisely those for which the vertices sharing edge
e have the same color. This color must be one of the wildcards {ck+1 , . . . , ck+` }, and so the
number of remaining proper (k, `)-colorings of G is counted by ` · χ(G/e)\v (k, `).
¤
With such a recurrence, we need to specify initial conditions. When G simply consists of
one vertex and has no edges, we have χG (x, y) = x + y, and when G is the empty graph,
we set χG (x, y) = 1 (consider G with one edge joining two vertices in (1.1)). Moreover, χ
is multiplicative on disconnected components. This allows us to compute χG for any graph
recursively.
In the special case when k = 1, there is also a way to calculate χG (1, y) by removing
vertices from G. Define the link of a vertex v to be the subgraph link(v) of G consisting of
v, the edges touching v, and the vertices sharing one of these edges with v. Also if u and v
are joined by an edge e, we define link(e) to be link(u) ∪ link(v) in G, and also we set deg(e)
to be deg(u) + deg(v) − 2. We then have the following rules.
Lemma 1.2. Let v be any vertex of G, and let e be any edge. Then,
χG (1, y) = y · χG\v (1, y) + y deg(v) · χG\link(v) (1, y),
(1.2)
χG\e (1, y) − y deg(e) · χG\link(e) (1, y).
(1.3)
χG (1, y) =
Proof. The number of proper (1, `)-colorings of G with vertex v colored with a wildcard is
` · χG\v (1, `). Moreover, in any proper coloring of G with v colored c1 , each vertex among the
deg(v) ones adjacent to v can only be one of the ` wildcards. This explains the first equality
in the lemma.
Let v be the vertex to which e contracts in G/e. From equation (1.2), we have
χG/e (1, y) = y · χ(G/e)\v (1, y) + y deg(v) · χ(G/e)\link(v) (1, y).
Subtracting this equation from (1.1) with x = 1, and noting that deg(e) = deg(v) and
G\link(e) = (G/e)\link(v), we arrive at the second equality in the lemma.
¤
Let Pn be the path graph on n vertices and let Cn be the cycle graph, also on n vertices
(C1 is a vertex with a loop attached while C2 is two vertices joined by two edges). Fixing
nonnegative integers k and `, not both zero, we define the following sequences of numbers
(n ≥ 1):
an = χPn (k, `),
bn = χCn (k, `).
(1.4)
As we shall see, these numbers are natural generalizations of both the Fibonacci and Lucas
numbers to the context of graph colorings. The following lemma uses graph decomposition
to give simple recurrences for these sequences.
AUGUST 2008/2009
221
THE FIBONACCI QUARTERLY
Lemma 1.3. The sequences an and bn satisfy the following linear recurrences with initial
conditions:
a1 = k + `,
a2 = (k + `)2 − k,
an = (k + ` − 1)an−1 + `an−2 ; (1.5)
b1 = `,
b2 = (k + `)2 − k,
b3 = a3 − b2 + `a1 ,
bn = (k + ` − 2)bn−1 + (k + 2` − 1)bn−2 + `bn−3 .
(1.6)
(1.7)
Moreover, the sequence bn satisfies a shorter recurrence if and only if k = 0, k = 1, or ` = 0.
When k = 0, this recurrence is given by bn = `bn−1 , and when k = 1, it is
bn = `bn−1 + `bn−2 .
(1.8)
Proof. The first recurrence follows from deleting an outer edge of the path graph Pn and
using Lemma 1.1. To verify the second one, we first use Lemma 1.1 (picking any edge in Cn )
to give
bn = an − bn−1 + `an−2 .
(1.9)
Let cn = bn + bn−1 = an + `an−2 and notice that cn satisfies the same recurrence as an ;
namely,
cn = an + `an−2
= (k + ` − 1)an−1 + `an−2 + ` ((k + ` − 1)an−3 + `an−4 )
= (k + ` − 1)(an−1 + `an−3 ) + `(an−2 + `an−4 )
(1.10)
= (k + ` − 1)cn−1 + `cn−2 .
It follows that bn satisfies the third order recurrence given in the statement of the lemma.
Additionally, the initial conditions for both sequences an and bn are easily worked out to be
the ones shown.
Finally, suppose that the sequence bn satisfies a shorter recurrence,
bn + rbn−1 + sbn−2 = 0,
and let


b3 b2 b1
B =  b4 b3 b2  .
b5 b4 b3
Since the nonzero vector [1, r, s]T is in the kernel of B, we must have that
0 = det(B) = −k 2 (k − 1)`((k + ` − 1)2 + 4`).
It follows that for bn to satisfy a smaller recurrence, we must have k = 0, k = 1, or ` = 0. It
is clear that when k = 0, we have bn = `n = `bn−1 . When k = 1, we can use Lemma 1.2 to
see that
bn+1 = `(an + `an−2 ),
and combining this with (1.9) gives the recurrence stated in the lemma.
¤
When k = 1 and ` = 1, the recurrences given by Lemma 1.3 when applied to the families
of path graphs and cycle graphs are the Fibonacci and Lucas numbers, respectively. This
observation is well-known (see [3, Examples 4.1 and 5.3]) and was brought to our attention
by Cox [1]:
χPn (1, 1) = Fn+2
and
χCn (1, 1) = Ln .
(1.11)
222
VOLUME 46/47, NUMBER 3
FIBONACCI IDENTITIES AND GRAPH COLORINGS
Moreover, when k = 2 and ` = 1, the recurrence given by Lemma 1.3 when applied to the
family of path graphs is the one associated to the Pell numbers:
χPn (2, 1) = Qn+1 ,
where Q0 = 1, Q1 = 1, and Qn = 2Qn−1 + Qn−2 .
2. Identities
In this section, we derive some identities involving the generalized Fibonacci and Lucas
numbers an and bn using the graph coloring interpretation found here. In what follows, we
fix k = 1. In this case, the an and bn satisfy the following recurrences:
an = `an−1 + `an−2
and
bn = `bn−1 + `bn−2 .
Theorem 2.1. The following identities hold:
bn = `an−1 + `2 an−3 ,
(2.1)
bn = an − `2 an−4 ,
(2.2)
2
(2.3)
ar+s = `ar as−1 + ` ar−1 as−2 ,
2
ar+s = ar as − ` ar−2 as−2 ,
(2.4)
ar+s+t+1 = `ar as at + `3 ar−1 as−1 at−1 − `4 ar−2 as−2 at−2 .
(2.5)
Proof. All the identities in the statement of the theorem follow from Lemma 1.2 when applied
to different graphs (with certain choices of vertices and edges). To see the first two equations,
consider the cycle graph Cn and pick any vertex and any edge. To see the next two equations,
consider the path graph Pr+s with v = r + 1 and e = {r, r + 1}.
s vertices
e
r vertices
v
t vertices
In order to prove the final equation in the statement of the theorem, consider the graph G
in the above figure. It follows from Lemma 1.2 that
`ar+s at + `3 ar−1 as−1 at−1 = ar+s+t+1 − `4 ar−2 as−2 at−1 .
Rearranging the terms and applying (2.4), we see that
ar+s+t+1 = `ar+s at + `3 ar−1 as−1 at−1 + `4 ar−2 as−2 at−1
= `(ar as − `2 ar−2 as−2 )at + `3 ar−1 as−1 at−1 + `4 ar−2 as−2 at−1
= `ar as at − `3 ar−2 as−2 (`at−1 + `at−2 )
+ `3 ar−1 as−1 at−1 + `4 ar−2 as−2 at−1
= `ar as at + `3 ar−1 as−1 at−1 − `4 ar−2 as−2 at−2 .
This completes the proof of the theorem.
AUGUST 2008/2009
¤
223
THE FIBONACCI QUARTERLY
Corollary 2.2. The following identities hold:
Ln = Fn+1 + Fn−1 ,
Ln = Fn+2 − Fn−2 ,
Fr+s = Fr+1 Fs + Fr Fs−1 ,
Fr+s = Fr+1 Fs+1 − Fr−1 Fs−1 ,
Fr+s+t = Fr+1 Fs+1 Ft+1 + Fr Fs Ft − Fr−1 Fs−1 Ft−1 .
Proof. The identities follow from the corresponding ones in Theorem 2.1 with ` = 1 by
making suitable shifts of the indices and using (1.11).
¤
3. Further Exploration
In this note, we have produced recurrences and identities by decomposing different classes
of graphs in different ways. Our treatment is by no means exhaustive, and there should be
many ways to expand on what we have done here. For instance, is there a graph coloring
proof of Cassini’s identity?
References
[1] D. Cox, Private communication, 2007.
[2] K. Dohmen, A. Pönitz, and P. Tittmann, A New Two-Variable Generalization of the Chromatic Polynomial, Discrete Math. Theor. Comput. Sci., 6.1 (2003) 69–89.
[3] T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (New York),
Wiley Interscience, NY, 2001.
MSC2000: 05C15, 05A19, 05C38
Department of Mathematics, Texas A&M University, College Station, TX 77843
E-mail address: [email protected]
Department of Mathematical Sciences, University of Copenhagen, Denmark
E-mail address: [email protected]
224
VOLUME 46/47, NUMBER 3